For a flow over a smooth sphere, does the wake region increases with increase in Reynolds number? And if so, then why?
[Physics] Effect of Reynolds number on wake region
flowfluid dynamicsturbulence
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Turbulence isn't the same as unsteadiness - a vortex street is not necessarily a turbulent phenomenon. As an analogy that (for some reason) I find easier to understand, consider a convection experiment where we heat a fluid at the bottom and cool it at the top. Below a certain threshold value for the temperature difference, the heat is transferred only by diffusion and there is no bulk flow. A little higher and we get an instability, resulting in the formation of a convection cell. In this case the fluid is moving, but it is still moving in a laminar way. As we increase the temperature difference, the speed of the flow increases, and it's only when we've increased the temperature quite a bit more that the flow becomes turbulent.
Vortex streets are similar. Above a certain value of the Reynolds number, the vortex street forms. The flow is now time-dependent, but it's periodic and still relatively easy to predict. If the flow is increased even further then the vortices spin so fast that smaller vortices form to dissipate their kinetic energy. It's only at this point that the flow becomes unpredictable and chaotic, which is when we call it turbulent. I guess you can say something like, a Kármán vortex street is a flow that's unsteady on one spatial scale, but in order for a flow to be called turbulent it has to be unsteady across a wide range of scales.
From the Wikipedia article for Reynolds number:
In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.
In addition to measuring the ratio of inertial to viscous forces in a flow, the incompressible Navier-Stokes equations can be written in non-dimensional form such that the only parameter is the Reynolds number (ignoring body forces). This is very nice because it is the basis for the validity of wind tunnel testing.
Suppose we would like to measure the aerodynamics of the flow around a Boeing 747. Two (at least) options exist:
- Build your very own full size 747, instrument it, and fly it. (extremely expensive)
- Build a small scale model of a 747, instrument it, test inside a wind tunnel (much less expensive)
But how do we know that the flow we measure in the wind tunnel is what really happens in flight? We match the Reynolds numbers and the exact same equations model both situations--therefore the aerodynamics must be the same. (Ignoring compressibility effects.)
Best Answer
Before the discussion of a sphere, I would like to mention how the flow across a long cylinder (i.e. a circle in 2 dimensions) progresses (and why so) with an increase in Reynolds number (Re).
Consider a flow across the cylinder in the creeping flow regime ($Re\leq 1$). This means that the inertial forces are low compared to the viscous forces. Consider what this implies. The inertial forces cause the fluid particles (and effectively the flow) to continue in its state of uniform motion. If the inertial forces are lower than the viscous forces, the viscous forces will dominate and the fluid particles do not continue in their state of uniform motion; which would be a tangent to the sphere at any point. Instead, they allow the viscous forces to manipulate the direction of motion, so that the fluid particles skirt along the curved surface of the cylinder. Thus in this regime, no wake is noticed and the flow is symmetric about the vertical diameter of the circle.
As the Re increases, the inertial forces start to increase causing the fluid to become more "stubborn" against the viscosity. This is primarily the reason for the flow to separate tangential to the cylinder. The point of separation depends on how dominant the inertial forces are as compared to the viscous forces; i.e. Re. With an increasing Re, the point of separation is drawn more upstream; causing larger wakes (which is quite easy to imagine).
This behavior continues till the point where the flow becomes turbulent. This is generally known as the "Drag-crisis"; where the drag-coefficient of the cylinder suddenly drops. Turbulent flow is more random in the sense that the momentum transfer in the fluid particles is randomly scattered in all directions; which causes the flow to lose some of its directional nature (which we earlier attributed to the inertial forces). Due to this, the viscous forces are assisted, causing the flow to reattach and get separated further downstream, resulting in a smaller wake in the process. The drag-crisis occurs at $Re=10^5 - 10^6$ for a long-cylinder.
After the drag-crisis, as Re is further increased, the separation begins to be drawn upstream again, causing an increase in the wake-size.
A similar reasoning will lead us to understand that the wake behind a sphere is definitely affected by the Re. And that too in a similar fashion. The increase in Re initially causes a growth in the size of the wake, and after transition to turbulence, a sudden reduction. This is again followed by an increase in the wake-size as Re is further increased.
I would strongly suggest you to check out An Album of Fluid Motion by Milton Van Dyke. It has interesting pictures both these flows, and many others, with good descriptions. It gives a good physical feel of the fluid flow in question.